Weakly sequentially complete Fr�chet spaces of integrable functions

Abstract

For a Frechet space E let \(L^1(\mu , E)\) be the space of all E-valued integrable functions with respect to a probability measure \(\mu \) and, when E is a real Frechet space, let \(L^1 ({\cal G}, E)\) be the space of all real-valued integrable functions with respect to an E-valued vector measure \({\cal G}\). We prove that if E is weakly sequentially complete then both \(L^1(\mu , E)\) and \(L^1({\cal G}, E)\) are weakly sequentially complete. We also prove that if E is a Frechet lattice, then E is weakly sequentially complete if and only if E does not contain any (lattice) copy of c0. By combining these main theorems, we also give some results about the existence of copies of c0 in \(L^1(\mu , E)\) and \(L^1({\cal G}, E)\).